# University of Chicago - Geometric Analysis Seminar

#### Department of Mathematics

We welcome all those who are interested to join us on Tuesdays, 1:00–2:00 PM, on zoom. To receive the zoom information please write an email to Daniel or Henrik.

### Spring 2021

#### April 13, 2021 Tue, 1:00–2:00PM

Salvatore Stuvard, UT Austin

The regularity of mass minimizing currents modulo p : Integer rectifiable currents mod(p) are a class of generalized surfaces in which it is possible to define and solve Plateau’s problem. The corresponding minimizers, mass minimizing currents mod(p), exhibit a far richer geometric complexity than the classical mass minimizing integral currents of Federer and Fleming. In this talk, I will present the partial interior regularity theory for these objects. The focus will be on dimension bounds and fine structural properties (such as rectifiability and local finiteness of measure) of their singular sets. The ultimate goal will be to reveal that singularities of mass minimizing currents mod(p) present an interesting “regular free boundary” structure. This is based on multiple joint works with Camillo De Lellis (IAS), Jonas Hirsch (U Leipzig), Andrea Marchese (U Trento), and Luca Spolaor (UCSD).

#### April 20, 2021 Tue, 1:00–2:00PM

Gabor Szekelyhidi, Notre Dame

Uniqueness of certain cylindrical tangent cones : Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over S^3 x S^3. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.

#### April 27, 2021 Tue, 1:00–2:00PM

Yi Lai, UC Berkeley

A family of 3d steady gradient Ricci solitons that are flying wings : We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

#### May 4, 2021 Tue, 1:00–2:00PM

Yangyang Li, Princeton

Generic Regularity of Minimal Hypersurfaces in Dimension 8 : The well-known Simons’ cone suggests that minimal hypersurfaces could be possibly singular in a Riemannian manifold with dimension greater than 7, unlike the low dimensional case. Nevertheless, it was conjectured that one could perturb away these singularities generically. In this talk, I will discuss how to perturb them away to obtain a smooth minimal hypersurface in an 8-dimension closed manifold, by induction on the "capacity" of singular sets. This result generalizes the previous works by N. Smale and by Chodosh-Liokumovich-Spolaor to any 8-dimensional closed manifold. This talk is based on joint work with Zhihan Wang.

#### May 11, 2021 Tue, 1:00–2:00PM

Davi Maximo, University of Pennsylvania

The Waist Inequality and Positive Scalar Curvature : The topology of three-manifolds with positive scalar curvature has been (mostly) known since the solution of the Poincare conjecture by Perelman. Indeed, they consist of connected sums of spherical space forms and S^2 x S^1's. In spite of this, their "shape" remains unknown and mysterious. Since a lower bound of scalar curvature can be preserved by a codimension two surgery, one may wonder about a description of the shape of such manifolds based on a codimension two data (in this case, 1-dimensional manifolds). In this talk, I will show results from a recent collaboration with Y. Liokumovich elucidating this question for closed three-manifolds

#### May 25, 2021 Tue, 1:00–2:00PM

Renato Bettiol, CUNY

Minimal 2-spheres in ellipsoids of revolution : Motivated by Morse-theoretic considerations, Yau asked in 1987 whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R^4 are planar, i.e., determined by the intersection with a hyperplane. Recently, this was shown not to be the case by Haslhofer and Ketover, who produced an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, combining Mean Curvature Flow and Min-Max methods. Using Bifurcation Theory and the symmetries that arise if at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with P. Piccione.

#### June 1, 2021 Tue, 10:00–11:00AM

Jaigyoung Choe, Korea Institute for Advanced Study

The periodic Plateau problem and its application : The periodic Plateau problem will be proposed and solved. As an application it will be proved that there exist four minimal annuli in a tetrahedron which are perpendicular to its faces. Also it will be proved that every Platonic solid contains three minimal surfaces of genus 0 perpendicular to its faces.

### Winter 2021

#### January 19, 2021 Tue, 1:00–2:00PM

Hans-Joachim Hein, Münster

Smooth asymptotics for collapsing Calabi-Yau metrics : Yau's solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion (locally uniformly away from the singular fibers) of Calabi-Yau metrics collapsing along a holomorphic fibration of a fixed compact Calabi-Yau manifold. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.

#### January 26, 2021 Tue, 1:00–2:00PM

Fritz Hiesmayr, UCL

An Urbano-type theorem for the Allen-Cahn equation : The Allen-Cahn equation is a semilinear elliptic PDE modelling phase transitions in two-phase media. In recent years this has found applications in geometry due to its link with minimal hypersurfaces. We present an analogue of Urbano's theorem about minimal surfaces in the round three-sphere. Our result is a rigidity theorem for solutions of the Allen-Cahn equation in S^3 with small index. They are symmetric, and vanish either on a minimal sphere or a Clifford torus. One key observation is that the nodal sets of two distinct solutions must have non-empty intersection.

#### February 2, 2021 Tue, 1:00–2:00PM

Daren Cheng, Waterloo

Existence of constant mean curvature 2-spheres in Riemannian 3-spheres : In this talk I’ll describe recent joint work with Xin Zhou, where we make progress on the question of finding closed constant mean curvature surfaces with controlled topology in 3-manifolds. We show that in a 3-sphere equipped with an arbitrary Riemannian metric, there exists a branched immersed 2-sphere with constant mean curvature H for almost every H. Moreover, the existence extends to all H when the target metric is positively curved. This latter result confirms, for the branched immersed case, a conjecture of Harold Rosenberg and Graham Smith.

#### February 9, 2021 Tue, 1:00–2:00PM

Aleksander Doan, Columbia

Counting pseudo-holomorphic curves in symplectic six-manifolds : The number of embedded pseudo-holomorphic curves in a symplectic manifold typically depends on the choice of an almost complex structure on the manifold and so does not lead to a symplectic invariant. However, I will discuss two instances in which such naive counting does define a symplectic invariant. The proof of invariance combines methods of symplectic geometry with results of geometric measure theory, especially regularity theory for calibrated currents. The talk is based on joint work with Thomas Walpuski. Time permitting, I will also discuss a related project, joint with Eleny Ionel and Thomas Walpuski, whose goal is to use geometric measure theory to prove the Gopakumar-Vafa finiteness conjecture.

#### February 16, 2021 Tue, 1:00–2:00PM

Costante Bellettini, UCL

Allen-Cahn minmax and multiplicity-1 minimal hypersurfaces : The existence of a closed minimal hypersurface in a compact Riemannian manifold was first established by the combined efforts of Almgren, Pitts, Schoen-Simon-Yau, Schoen-Simon in the early 80s by means of what is nowadays called minmax a la Almgren-Pitts. An alternative approach to reach the same existence result has been implemented in recent years in a work by Guaraco, using a minmax construction for the Allen-Cahn energy, in combination with works by Hutchinson-Tonegawa, Tonegawa, Tonegawa-Wickramasekera, Wickramasekera. A natural question (ubiquitous in geometric analysis and, in particular, in minmax constructions) is whether the minimal hypersurface is obtained with multiplicity 1. The multiplicity-1 information has important geometric consequences. However, the a priori possibility of higher multiplicity is intrinsic in both minmax constructions, as they are carried out in the class of varifolds. After an overview, this talk focuses on the case of an ambient Riemannian manifold (of dimension 3 or higher) with positive Ricci curvature: in this case, the minmax construction via Allen-Cahn yields a multiplicity-1 minimal hypersurface. If time permits, the case of low-dimensional manifolds endowed with a bumpy metric will also be addressed.

#### February 23, 2021 Tue, 1:00–2:00PM

Robin Neumayer, Northwestern

#### March 7, 2017 Tue, 3:00–4:00PM, Ryerson 358

Victoria Sadovskaya, Department of Mathematics - Penn State

Boundedness, compactness, and invariant norms for operator-valued cocycles.: We consider group-valued cocycles over dynamical systems with hyperbolic behavior, such as hyperbolic diffeomorphisms or subshifts of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Holder continuous. We consider the periodic data of A, i.e. the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Holder continuous family of norms. This is joint work with B. Kalinin.

#### Mar 14, 2017 Tue, 3:00–4:00PM, Ryerson 358

Baris Coskunuser , Boston College

Embeddedness of the solutions to the H-Plateau Problem : In this talk, we will give a generalization of Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. arXiv:1504.00661

### FALL 2016

#### Oct 11, 2016 Tue, 3:00–4:00PM, Eckhart 207

Lu Wang, Department of Mathematics - University of Wisconsin-Madison

Asymptotic structure of self-shrinkers: We show that each end of a noncompact self-shrinker in Euclidean 3-space of finite topology is smoothly asymptotic to a regular cone or a self-shrinking round cylinder.

#### Oct 18, 2016 Tue, 3:00–4:00PM, Eckhart 207

Spencer Becker-Kahn, Department of Mathematics - MIT

Zero Sets of Smooth Functions and p-Sweepouts: It is well known that any closed set can occur as the zero set of a smooth function. But in many contexts in analysis, one does not work with arbitrary smooth functions; one works with smooth functions that vanish to only finite order. And the zero set of any such function must have some regularity and structure. With recourse to an analogy made by Gromov and some recent conjectures of Marques and Neves, I will explain the application of some general results about such zero sets to p-sweepouts (which are p-dimensional families of generalized hypersurfaces that sweepout a smooth manifold in a certain sense). In the course of doing so, I will explain a new result about the regularity of zero sets of smooth functions near a point of finite vanishing order (joint work with Tom Beck and Boris Hanin)..

#### Nov 8, 2016 Tue, 3:00–4:00PM, Eckhart 207 There will be two seminars on this date

Jonathan Zhu, Department of Mathematics - Harvard University

Entropy and self-shrinkers of the mean curvature flow: The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.

#### Nov 8, 2016 Tue, 4:00–5:00PM, Eckhart 207 There will be two seminars on this date

Greg Chambers, Department of Mathematics - The University of Chicago

Existence of minimal hypersurfaces in non-compact manifolds: I prove that every complete non-compact manifold of finite volume admits a minimal hypersurface of finite area. This work is in collaboration with Yevgeny Liokumovich.

#### Nov 15, 2016 Tue, 3:00–4:00PM, Eckhart 207

Peter McGrath, Brown University

New doubling constructions for minimal surfaces: I will discuss recent work (with Nikolaos Kapouleas) on constructions of new embedded minimal surfaces using singular perturbation 'gluing' methods. I will discuss at length doublings of the equatorial S^2 in S^3. In contrast to earlier work of Kapouleas, the catenoidal bridges may be placed on arbitrarily many parallel circles on the base S^2. This necessitates a more detailed understanding of the linearized problem and more exacting estimates on associated linearized doubling solutions.

#### Nov 22, 2016 Tue, 3:00–4:00PM, Eckhart 207 There will be two seminars on this date

Franco Vargas Pallete, UC Berkeley

Renormalized volume: The renormalized volume V_R is a finite quantity associated to certain hyperbolic 3-manifolds of infinite volume. In this talk I'll discuss its definition and some properties for acylindrical manifolds, namely local convexity and convergence under geometric limits. If time suffices I'll also discuss some applications and further examples.

#### Nov 22, 2016 Tue, 4:00–5:00PM, Eckhart 207 There will be two seminars on this date

Christos Mantoulidis, Stanford University

Fill-ins, extensions, scalar curvature, and quasilocal mass: There is a special relationship between the Jacobi and the ambient scalar curvature operator. I'll talk about an extremal bending technique that exploits this relationship. It lets us compute the Bartnik mass of apparent horizons and disprove a form of the Hoop conjecture due to G. Gibbons. Then, I'll talk about a derived "cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. It is used in studying a priori L^1 estimates for boundary mean curvature for compact initial data sets, and in generalizing Brown-York mass. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.

#### Nov 29, 2016 Tue, 3:00–4:00PM, Eckhart 207

Hung Tran, UC Irvine

Index Characterization for Free Boundary Minimal Surfaces: A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.

For further information on this seminar or this webpage, please contact Marco Guaraco at guaraco(at)math.uchicago.edu, Lucas Ambrozio at lambrozio(at)math.uchicago.edu or Rafael Montezuma at montezuma(at)math.uchicago.edu